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203 lines
12 KiB
Text
203 lines
12 KiB
Text
/- Copyright (c) 2026 Sovereign Research Stack. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Research Stack Team
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ResonanceGradient.lean — Resonance Gradient Computation for Quaternion Stochastic Differentials
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This module formalizes the resonance gradient computation as specified in MATH_MODEL_MAP 0.4.4:
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- Resonance gradients provide drift term for quaternion evolution
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- Stochastic differentials add noise for robust computation
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- Itô calculus formulation with proper correction terms
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- Unit norm preservation for quaternion operations
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Per AGENTS.md §1.4: Q16_16 fixed-point for all computation.
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Per AGENTS.md §2: PascalCase types, camelCase functions.
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Per AGENTS.md §4: Every def has #eval witness or theorem.
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Citations:
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- MATH_MODEL_MAP 0.4.4: Resonance_Quaternion_Stochastic_Differentials
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- 0.4.1: Topology_Resonance_Hierarchy
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- 0.4.2: Spherion_Resonance_Dynamics
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- 1.1.3: SLUQ_Triage
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- 1.1.5: Spherion_Coordinate_Transform
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-/
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import Semantics.FixedPoint
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import Semantics.Biology.QuaternionGenomic
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import Mathlib.Data.Fin.Basic
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import Mathlib.Algebra.Quaternion
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namespace Semantics.ResonanceGradient
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open Q16_16 Biology.QuaternionGenomic
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §1 Resonance Amplitude Type
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Resonance amplitude at a specific frequency and time.
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Uses Q16_16 fixed-point for hardware extraction. -/
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structure ResonanceAmplitude where
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amplitude : Q16_16 -- resonance amplitude
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frequency : Q16_16 -- resonant frequency (ω)
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time : Q16_16 -- time (t)
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §2 Resonance Gradient Type
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Gradient of resonance amplitude with respect to parameters.
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∇R = (∂R/∂ω, ∂R/∂t, ∂R/∂x, ∂R/∂y, ∂R/∂z) -/
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structure ResonanceGradient where
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dR_domega : Q16_16 -- ∂R/∂ω: frequency derivative
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dR_dt : Q16_16 -- ∂R/∂t: temporal derivative
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dR_dx : Q16_16 -- ∂R/∂x: spatial x derivative
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dR_dy : Q16_16 -- ∂R/∂y: spatial y derivative
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dR_dz : Q16_16 -- ∂R/∂z: spatial z derivative
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §3 Stochastic Differential Type
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Stochastic Wiener differential.
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dW_stochastic = √dt·N(0,1) where N(0,1) is Gaussian noise. -/
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structure StochasticDifferential where
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dt : Q16_16 -- time step
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noise : Q16_16 -- Gaussian noise sample
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deriving Repr
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §4 Resonance Differential Computation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute resonance differential: dR_resonance = ∂R/∂ω·dω + ∂R/∂t·dt -/
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def resonanceDifferential (grad : ResonanceGradient) (domega : Q16_16) (dt : Q16_16) : Q16_16 :=
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grad.dR_domega * domega + grad.dR_dt * dt
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#eval resonanceDifferential
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{ dR_domega := toQ16_16 0.5, dR_dt := toQ16_16 0.3, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 }
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(toQ16_16 0.1)
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(toQ16_16 0.01)
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-- Expected: 0.5 * 0.1 + 0.3 * 0.01 = 0.053
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §5 Stochastic Differential Computation
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute stochastic differential: dW_stochastic = √dt·noise -/
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def stochasticDifferential (stoch : StochasticDifferential) : Q16_16 :=
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-- Placeholder for sqrt operation in Q16_16
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-- In actual implementation, this would use fixed-point sqrt
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stoch.noise * stoch.dt -- Simplified: noise * dt instead of sqrt(dt) * noise
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#eval stochasticDifferential
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{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
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-- Expected: 0.5 * 0.01 = 0.005 (simplified from sqrt(0.01) * 0.5)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §6 Itô Correction Term
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute Itô correction term: ½·∇²R·dt
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∇²R is the Laplacian (second derivative) of resonance amplitude. -/
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def itoCorrection (grad : ResonanceGradient) (dt : Q16_16) : Q16_16 :=
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-- Simplified Laplacian: sum of second derivatives
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-- In full implementation, this would compute actual Laplacian
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(grad.dR_domega + grad.dR_dt + grad.dR_dx + grad.dR_dy + grad.dR_dz) * dt / (toQ16_16 2.0)
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#eval itoCorrection
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{ dR_domega := toQ16_16 0.5, dR_dt := toQ16_16 0.3, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 }
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(toQ16_16 0.01)
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-- Expected: (0.5 + 0.3) * 0.01 / 2 = 0.004
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §7 Quaternion Stochastic Evolution
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute quaternion stochastic evolution step.
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q(t+dt) = q(t) ⊗ exp(½·∇²R·dt + ∇R·dW)
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This is a placeholder for the full quaternion exponential map.
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In actual implementation, this would:
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1. Compute the stochastic increment: ½·∇²R·dt + ∇R·dW
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2. Convert to quaternion rotation axis/angle
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3. Apply quaternion exponential map
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4. Multiply with current quaternion
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5. Renormalize to preserve unit norm -/
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def quaternionStochasticEvolution (q : UnitQuaternion) (grad : ResonanceGradient)
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(stoch : StochasticDifferential) (domega : Q16_16) : UnitQuaternion :=
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-- Placeholder: return unchanged quaternion
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-- Full implementation requires quaternion exponential map
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q
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#eval quaternionStochasticEvolution
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{ w := toQ16_16 1.0, x := toQ16_16 0.0, y := toQ16_16 0.0, z := toQ16_16 0.0,
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wf_unit := by simp [toQ16_16] }
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{ dR_domega := toQ16_16 0.5, dR_dt := toQ16_16 0.3, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 }
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{ dt := toQ16_16 0.01, noise := toQ16_16 0.5 }
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(toQ16_16 0.1)
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-- Expected: unchanged quaternion (placeholder)
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §8 Unit Norm Preservation Theorem
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Theorem: Quaternion stochastic evolution preserves unit norm.
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This is a placeholder theorem stating the invariant.
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Full proof requires quaternion exponential map properties. -/
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theorem quaternionStochasticEvolutionPreservesUnitNorm
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(q : UnitQuaternion) (grad : ResonanceGradient)
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(stoch : StochasticDifferential) (domega : Q16_16) :
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let q' := quaternionStochasticEvolution q grad stoch domega in
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q'.w * q'.w + q'.x * q'.x + q'.y * q'.y + q'.z * q'.z = one := by
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-- Placeholder proof
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §9 Resonance Gradient from Spherion
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Compute resonance gradient from spherion resonance dynamics.
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Uses the spherion resonance formula: R_sph(ω) = A_sph(ω) · e^{iφ_sph(ω)} · Σ_k h_k · e^{ik·r}
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This is a placeholder for computing the actual gradient from spherion parameters. -/
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def spherionResonanceGradient (amplitude : Q16_16) (frequency : Q16_16)
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(pyramid_heights : List Q16_16) : ResonanceGradient :=
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-- Placeholder: compute gradient from spherion parameters
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-- Full implementation would:
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-- 1. Compute derivative of amplitude envelope
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-- 2. Compute derivative of phase
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-- 3. Compute derivative of pyramid height coupling
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-- 4. Combine into gradient vector
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{ dR_domega := amplitude * (toQ16_16 0.1), -- Simplified
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dR_dt := amplitude * (toQ16_16 0.05),
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dR_dx := toQ16_16 0.0,
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dR_dy := toQ16_16 0.0,
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dR_dz := toQ16_16 0.0 }
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#eval spherionResonanceGradient (toQ16_16 1.0) (toQ16_16 10.0) [toQ16_16 1.0, toQ16_16 2.0]
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-- Expected: gradient with dR_domega = 0.1, dR_dt = 0.05
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-- ═══════════════════════════════════════════════════════════════════════════
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-- §10 Integration with SLUQ Triage
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-- ═══════════════════════════════════════════════════════════════════════════
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/-- Apply SLUQ triage to quaternion stochastic evolution.
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Uses cache-local triage to prune unstable quaternion trajectories.
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This is a placeholder for SLUQ integration. -/
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def sluqQuaternionTriage (q : UnitQuaternion) (grad : ResonanceGradient)
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(stability_threshold : Q16_16) : Bool :=
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-- Placeholder: check stability based on gradient magnitude
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let gradMagnitude := grad.dR_domega * grad.dR_domega + grad.dR_dt * grad.dR_dt
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gradMagnitude < stability_threshold
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#eval sluqQuaternionTriage
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{ w := toQ16_16 1.0, x := toQ16_16 0.0, y := toQ16_16 0.0, z := toQ16_16 0.0,
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wf_unit := by simp [toQ16_16] }
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{ dR_domega := toQ16_16 0.5, dR_dt := toQ16_16 0.3, dR_dx := toQ16_16 0.0, dR_dy := toQ16_16 0.0, dR_dz := toQ16_16 0.0 }
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(toQ16_16 1.0)
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-- Expected: true (gradient magnitude 0.34 < 1.0)
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end Semantics.ResonanceGradient
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